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Separated out matrix_space examples
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@emikelsons
emikelsons committed Feb 6, 2025
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158 changes: 48 additions & 110 deletions docs/src/matrix.md
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Expand Up @@ -11,9 +11,8 @@ As well as the Ring and Matrix interfaces, the following functions are provided
manipulate matrices and to set and retrieve entries and other basic data associated
with the matrices.


It is possible to create matrices directly, without first
creating the corresponding matrix space (the inner constructor being called directly).
creating a corresponding matrix space.

```julia
matrix(R::Ring, arr::Matrix{T}) where T <: RingElement
Expand Down Expand Up @@ -54,14 +53,6 @@ julia> P = zero_matrix(ZZ, 3, 2)
[0 0]
[0 0]
[0 0]
julia> R = matrix_ring(ZZ, 2)
Matrix ring of degree 2
over integers
julia> M = R()
[0 0]
[0 0]
```

```@docs
Expand Down Expand Up @@ -94,56 +85,34 @@ Base.map!(f, ::MatrixElem{S}, ::MatrixElem{T}) where {S <: RingElement, T <: Rin

## Inverse

```@docs
```@docs; canonical=false
Base.inv{T <: RingElement}(::MatrixElem{T})
is_invertible_with_inverse{T <: RingElement}(::MatrixElem{T})
is_invertible{T <: RingElement}(::MatrixElem{T})
is_invertible_with_inverse{T <: RingElement}(::MatrixElem{T})
pseudo_inv(M::MatrixElem{T}) where {T <: RingElement}
```

**Examples**

```jldoctest
julia> R, x = polynomial_ring(QQ, :x)
(Univariate polynomial ring in x over rationals, x)
```@jldoctest
julia> M = matrix(QQ, 3, 3, [1 2 3;4 5 6;0 0 1])
[1//1 2//1 3//1]
[4//1 5//1 6//1]
[0//1 0//1 1//1]
julia> K, = residue_field(R, x^3 + 3x + 1); a = K(x);
julia> X = inv(M)
[-5//3 2//3 1//1]
[ 4//3 -1//3 -2//1]
[ 0//1 0//1 1//1]
julia> S = matrix_space(K, 3, 3)
Matrix space of 3 rows and 3 columns
over residue field of R modulo x^3 + 3*x + 1
julia> A = S([K(0) 2a + 3 a^2 + 1; a^2 - 2 a - 1 2a; a^2 + 3a + 1 2a K(1)])
[ 0 2*x + 3 x^2 + 1]
[ x^2 - 2 x - 1 2*x]
[x^2 + 3*x + 1 2*x 1]
julia> X = inv(A)
[-343//7817*x^2 + 717//7817*x - 2072//7817 -4964//23451*x^2 + 2195//23451*x - 11162//23451 -232//23451*x^2 - 4187//23451*x - 1561//23451]
[ 128//7817*x^2 - 655//7817*x + 2209//7817 599//23451*x^2 - 2027//23451*x - 1327//23451 -1805//23451*x^2 + 2702//23451*x - 7394//23451]
[ 545//7817*x^2 + 570//7817*x + 2016//7817 -1297//23451*x^2 - 5516//23451*x - 337//23451 8254//23451*x^2 - 2053//23451*x + 16519//23451]
julia> is_invertible(A)
julia> is_invertible(M)
true
julia> is_invertible_with_inverse(A)
(true, [-343//7817*x^2+717//7817*x-2072//7817 -4964//23451*x^2+2195//23451*x-11162//23451 -232//23451*x^2-4187//23451*x-1561//23451; 128//7817*x^2-655//7817*x+2209//7817 599//23451*x^2-2027//23451*x-1327//23451 -1805//23451*x^2+2702//23451*x-7394//23451; 545//7817*x^2+570//7817*x+2016//7817 -1297//23451*x^2-5516//23451*x-337//23451 8254//23451*x^2-2053//23451*x+16519//23451])
julia> R, x = polynomial_ring(ZZ, :x)
(Univariate polynomial ring in x over integers, x)
julia> S = matrix_space(R, 3, 3)
Matrix space of 3 rows and 3 columns
over univariate polynomial ring in x over integers
julia> is_invertible_with_inverse(M)
(true, [-5//3 2//3 1; 4//3 -1//3 -2; 0 0 1])
julia> A = S([R(0) 2x + 3 x^2 + 1; x^2 - 2 x - 1 2x; x^2 + 3x + 1 2x R(1)])
[ 0 2*x + 3 x^2 + 1]
[ x^2 - 2 x - 1 2*x]
[x^2 + 3*x + 1 2*x 1]
julia> X, d = pseudo_inv(A)
([4*x^2-x+1 -2*x^3+3 x^3-5*x^2-5*x-1; -2*x^3-5*x^2-2*x-2 x^4+3*x^3+2*x^2+3*x+1 -x^4+x^2+2; -x^3+2*x^2+2*x-1 -2*x^3-9*x^2-11*x-3 2*x^3+3*x^2-4*x-6], -x^5 - 2*x^4 - 15*x^3 - 18*x^2 - 8*x - 7)
julia> pseudo_inv(M)
([5 -2 -3; -4 1 6; 0 0 -3], -3//1)
```

## Submatrices
Expand Down Expand Up @@ -176,7 +145,6 @@ julia> N2 = M[:, :]
julia> N3 = M[2:3, 2:3]
[3 4]
[4 5]
```

As per Julia, AbstractAlgebra supports the construction of matrix views.
Expand Down Expand Up @@ -422,40 +390,30 @@ Rationals

## LU factorisation

```@docs
```@docs; canonical=false
lu{T <: FieldElem}(::MatElem{T}, ::SymmetricGroup)
fflu{T <: RingElem}(::MatElem{T}, ::SymmetricGroup)
```

**Examples**

```jldoctest
julia> R, x = polynomial_ring(QQ, :x)
(Univariate polynomial ring in x over rationals, x)
julia> K, = residue_field(R, x^3 + 3x + 1); a = K(x);
```@jldoctest
julia> M = matrix(QQ, 3, 3, [1 2 3;4 5 6;0 0 1])
[1//1 2//1 3//1]
[4//1 5//1 6//1]
[0//1 0//1 1//1]
julia> S = matrix_space(K, 3, 3)
Matrix space of 3 rows and 3 columns
over residue field of R modulo x^3 + 3*x + 1
julia> A = S([K(0) 2a + 3 a^2 + 1; a^2 - 2 a - 1 2a; a^2 - 2 a - 1 2a])
[ 0 2*x + 3 x^2 + 1]
[x^2 - 2 x - 1 2*x]
[x^2 - 2 x - 1 2*x]
julia> r, P, L, U = lu(A)
(2, (1,2), [1 0 0; 0 1 0; 1 0 1], [x^2-2 x-1 2*x; 0 2*x+3 x^2+1; 0 0 0])
julia> r, d, P, L, U = fflu(A)
(2, 3*x^2 - 10*x - 8, (1,2), [x^2-2 0 0; 0 3*x^2-10*x-8 0; x^2-2 0 1], [x^2-2 x-1 2*x; 0 3*x^2-10*x-8 -4*x^2-x-2; 0 0 0])
julia> r, P, L, U = lu(M)
(3, (), [1 0 0; 4 1 0; 0 0 1], [1 2 3; 0 -3 -6; 0 0 1])
julia> r, d, P, L, U = fflu(M)
(3, -3//1, (), [1 0 0; 4 -3 0; 0 0 -3], [1 2 3; 0 -3 -6; 0 0 -3])
```

## Reduced row-echelon form

```@docs
```@docs; canonical=false
rref_rational{T <: RingElem}(::MatElem{T})
rref{T <: FieldElem}(::MatElem{T})
Expand All @@ -465,46 +423,32 @@ is_rref{T <: FieldElem}(::MatElem{T})

**Examples**

```jldoctest
julia> R, x = polynomial_ring(QQ, :x)
(Univariate polynomial ring in x over rationals, x)
julia> K, = residue_field(R, x^3 + 3x + 1); a = K(x);
```@jldoctest
julia> M = matrix(QQ, 3, 3, [1 2 3;4 5 6;0 0 1])
[1//1 2//1 3//1]
[4//1 5//1 6//1]
[0//1 0//1 1//1]
julia> S = matrix_space(K, 3, 3)
Matrix space of 3 rows and 3 columns
over residue field of R modulo x^3 + 3*x + 1
julia> r1, A = rref(M)
(3, [1 0 0; 0 1 0; 0 0 1])
julia> M = S([K(0) 2a + 3 a^2 + 1; a^2 - 2 a - 1 2a; a^2 + 3a + 1 2a K(1)])
[ 0 2*x + 3 x^2 + 1]
[ x^2 - 2 x - 1 2*x]
[x^2 + 3*x + 1 2*x 1]
julia> N = matrix(ZZ, 3, 3, [1 2 3;4 5 6;0 0 1])
[1 2 3]
[4 5 6]
[0 0 1]
julia> r, A = rref(M)
(3, [1 0 0; 0 1 0; 0 0 1])
julia> r2, B = rref_rational(N)
(3, [-3 0 0; 0 -3 0; 0 0 -3], -3)
julia> is_rref(A)
true
julia> R, x = polynomial_ring(ZZ, :x)
(Univariate polynomial ring in x over integers, x)
julia> S = matrix_space(R, 3, 3)
Matrix space of 3 rows and 3 columns
over univariate polynomial ring in x over integers
julia> M = S([R(0) 2x + 3 x^2 + 1; x^2 - 2 x - 1 2x; x^2 + 3x + 1 2x R(1)])
[ 0 2*x + 3 x^2 + 1]
[ x^2 - 2 x - 1 2*x]
[x^2 + 3*x + 1 2*x 1]
julia> r, A, d = rref_rational(M)
(3, [-x^5-2*x^4-15*x^3-18*x^2-8*x-7 0 0; 0 -x^5-2*x^4-15*x^3-18*x^2-8*x-7 0; 0 0 -x^5-2*x^4-15*x^3-18*x^2-8*x-7], -x^5 - 2*x^4 - 15*x^3 - 18*x^2 - 8*x - 7)
julia> is_rref(A)
julia> is_rref(B)
true
```

## Other functionality

### Symmetry testing

```@docs
Expand Down Expand Up @@ -596,23 +540,18 @@ nullspace{T <: FieldElem}(::MatElem{T})

### Hessenberg form

```@docs
```@docs; canonical=false
hessenberg{T <: RingElem}(::MatElem{T})
is_hessenberg{T <: RingElem}(::MatElem{T})
```

**Examples**

```jldoctest
```@jldoctest
julia> R, = residue_ring(ZZ, 7);
julia> S = matrix_space(R, 4, 4)
Matrix space of 4 rows and 4 columns
over residue ring of integers modulo 7
julia> M = S([R(1) R(2) R(4) R(3); R(2) R(5) R(1) R(0);
R(6) R(1) R(3) R(2); R(1) R(1) R(3) R(5)])
julia> M = matrix(R, 4, 4, [1 2 4 3; 2 5 1 0;6 1 3 2; 1 1 3 5])
[1 2 4 3]
[2 5 1 0]
[6 1 3 2]
Expand All @@ -626,7 +565,6 @@ julia> A = hessenberg(M)
julia> is_hessenberg(A)
true
```

### Characteristic polynomial
Expand Down
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